A Marginal-cost Pricing Model for Transportation Networks

A Marginal-cost Pricing Model for Transportation Networks with

Multiple-class Users – An Application to the Toll Ceiling Problem

Shou-Ren HU a, Hui Pei HUANG b

a Department of Transportation and Communication Management Science, National

Cheng Kung University, Tainan City, 70101, Taiwan a E-mail: [email protected] b THI Consultants lnc., Taichung City, 40466, Taiwan b E-mail: [email protected]

Abstract: This study proposes a marginal-cost pricing model for the congestion pricing

problem under different users’ attributes and their influence on the system’s total cost. The

primary objective is to investigate the effects of the current double tolling policy during

congested periods of the national freeway system in Taiwan. The proposed model solves the

differential pricing problem by the marginal cost pricing theory under different user

characteristics with varying external costs and values of time. The results of the numerical

analysis indicate that the toll ceiling by law needs to be relaxed so that the ultimate goal of

social welfare maximization can be achieved, and the marginal-cost pricing strategies have an

effect on reducing traffic congestion by encouraging users with a high price elasticity to

switch to alternative routes. The results found in this study should have implications for the

government offices in preparing a desirable tolling policy.

Keywords: Marginal-cost Pricing, Multiple-class Users, Asymmetric Link Cost, External Cost,

Value of Time

1. INTRODUCTION

As the increasing economic and social activities, traffic demand and usage of private vehicles

have been increasing in the metropolitan areas. The growth and increased usage of private

vehicles have resulted in many problems, such as traffic congestion, road crashes, and air and

noise pollutions. To resolve these problems, traditional and advanced methods have been

proposed, in which congestion pricing and/or highway tolling policy has been shown as one

of the most effective policy tools to mitigate traffic congestion problems. In Taiwan, a

distance-based Electronic Toll Collection (ETC) system was implemented on the national

freeway system since 2013. Averagely four millions of toll transactions in a typical weekday.

The distance-based ETC system uses a microwave system for the communication between an

on-board RFID tag (called e-Tag, free-of-charge of installation) and a roadside gantry. Toll is

charged in two ways. For those e-Tag equipped vehicles, they are charged by a pre-registered

deposit account. While for those vehicles without an e-Tag, they will receive a bill via

traditional mail services. The tolls for non e-Tag users are calculated by their travel distance

captured/recorded by a license plate recognition (LPR) system plus an administrative fee.

Totally 319 gantries are installed near an entrance and or an exit for each mainline segment.

The average distance between two gantries is 3.13 kilometers. The ETC system does not only

provide the tolling service, but also has the capability of conducting differential pricing

strategies according to different vehicle types and/or user classes. Besides the tolling function,

the ETC system also provides time-dependent vehicular origin-destination (O-D) data, travel

Journal of the Eastern Asia Society for Transportation Studies, Vol.12, 2017

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time and distance data at the individual vehicle level. The valuable traffic data can be used for

effective freeway control and management.

For the toll levels of the national freeway system in Taiwan, the standard toll structure set

by Ministry of Transportation and Communications (MOTC) is shown in Table 1. Differential

tolling policy have been separately implemented on specific periods. For instance, 150% and

50% of the standard toll are respectively imposed on the vehicles traveling on the peak and

off-peak hours of series holidays. Despite for that, a freeway corridor connecting major

metropolitan or attraction areas is frequently congested during specific time intervals and road

segments. This outcome raises a question if the upper bound of the differential toll rates that is

set two times of the standard toll levels (i.e. the ceiling by law) is not high enough to prevent

the traveling public from using the limited freeway resource. Thereby, the purpose of this

study is to investigate the toll rates for a social optimal situation by charging different user

classes with differential tolls. The ultimate goal of this study is to provide the government

agency and freeway bureau a reference in setting a desirable toll structure.

Table 1. Toll structure of the national freeway system in Taiwan

Unit: NT$/km

Source: Taiwan Area National Freeway Bureau (TANFB), MOTC, 2015.

Road pricing is an effective method to mitigate traffic congestion problem through

changing road users’ route choice behaviors. To deal with the congestion pricing problem, the

policy of road pricing has been implemented in several cities around the world (Palma and

Lindsey, 2011) and the theoretical researches of road pricing has been widely extended and

discussed under different charging schemes since Pigou (1920) first advanced the concept of

road pricing of congestion tax theory. The theory proposed by Pigou is named marginal-cost

pricing principle. The ultimate goal of the marginal-cost pricing theory is to internalize the

external cost in order to achieve the social optimum objective and remove and/or divert the

excessive traffic flow or demand to alternative routes (Pigou, 1920; 3. Knight, 1924; Walters,

1961; Vickrey, 1969). By imposing the extra cost on the specific users, the limited highway

resources are expected to be used in a more efficient manner, and the traffic congestion

problem can be mitigated.

Assumption on homogeneity of highway users, no matter on vehicle types or values of

time (VOTs), does enormously simplify the theoretical development and the empirical study,

but this assumption is unrealistic (Walters, 1961). In order to reflect the reality, it needs to

account for different travel costs and VOTs when solving the congestion pricing problem. The

different user classes (e.g., passenger car, truck, bus) have different characteristics (e.g.

income, trip purpose, vehicle type) and are associated with varying VOTs (Arnott, 1992;

Small and Yan, 2001; Yang and Huang, 2005; Holguín-Veras and Cetin, 2009), resulting in

different levels of externality (Arnott, 1992; Holguín-Veras and Cetin, 2009; Dafermos, 1972;

Smith, 1979; Chen and Bernstein, 2004). The time value will influence users’ route choice

behaviors. Different types of vehicle will induce different external costs due to vehicle size,

acceleration/dissertation characteristics and so on (Chen and Bernstein, 2004), and should be

charged by differential tolls. Although the issue of road pricing with heterogeneous users have

Travel distance (d)

Vehicle type

d km/day 20 km < d 200 km d 200 km

Passenger car 0 1.2 0.9

Heavy vehicle 0 1.5 1.12

Trailer 0 1.8 1.35


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been investigated by the past researches from the perspectives of different travel costs or

VOTs, relatively few studies examine different external costs and VOTs together to solve the

congestion pricing problem. This topic is worthy of investigation, especially from a modern

highway management’s perspective.

Solving the congestion pricing problem with heterogeneous users is complex in a

general network. Dafermos (1972) suggested that the multiple-class equilibrium model could

simplify to a single-class model with link flow interaction by copying the network with

number of vehicle types to express the interaction of different vehicle types traveling on the

same link. To solve the network equilibrium problem with asymmetric link cost function, the

diagonalization method was adopted; this method is as the common solution algorithm for the

asymmetric (cost) network equilibrium problem (Shiffi, 1985). The congestion pricing

problem with multiple vehicle types or user classes can be solved by this method (Yang and

Huang, 2005).

Accordingly, the primary objective of this study are to evaluate congestion tolls of

different vehicle types by accounting for the different external costs and VOTs associated with

different user classes based on a system optimum criterion. We construct a marginal-cost

congestion pricing model based on the national freeway network in Taiwan to explore a

reasonable ceiling for the current tolling structure implemented in Taiwan.

2. METHODOLGY

The road pricing model is formulated by marginal-cost pricing theory and the solution

algorithm is developed to solve the marginal-cost pricing problem on a general transportation

network with multiple-class users.

2.1 Model Assumptions

Before constructing the congestion pricing models, assumptions are made in the following.

First, the corresponding link travel time cost function is a strictly increasing, convex and

continuously differentiable function with the increasing flow on link, and the corresponding

O-D demand function is nonnegative and strictly decreasing with respect to the generalized

cost for each class m∈M between an O-D pair. Second, the route choice behaviors of road

users follow the user equilibrium principle, which means that no user can improve his/her

travel time by unilaterally changing, route. Under this principle, each road user is perfectly

rational to choose cost-minimizing paths between any O-D pair. Third, road users are

heterogeneous and classified by two kinds of vehicle types, which are cars and heavy vehicles.

The road users of class m have their own cost function and value of time. Forth, the potential

demand of an O-D pair r-s will not alter in a short time period. Fifth, congestion tolls can be

charged on all links. Sixth, the externalities other than traffic congestion are not considered.

Seventh, the roadway capacity is fixed.

2.2 Pricing Model

This study formulates the model of a system optimum condition to discuss the congestion toll

with heterogeneous users whose cost functions are asymmetric and value of time are varied

on a general transportation network. The cost function is formulated as a travel time function

which is not only affected by the flows associated with its own vehicle type (called the main

effect) but also influenced by those of other vehicles (called cross effect).This study uses


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Bureau of Public Roads (BPR) function as the cost function (Yang and Huang, 2005; Chen and

Bernstein, 2004). For each link , there is a travel time cost function of

for each vehicle type (passenger car and heavy vehicle):

(1a)

(1b)

where is traffic flow of class on link ; is travel time cost function of

; and are the parameter of the BPR function of class ; is capacity of

class on link ; E is the passenger car equivalence of heavy vehicle.

Generally, each vehicle type can drive on every lane and the capacities of different

vehicle types are the same. However, under certain situations (e.g. HOV lane), the capacities

of different vehicle types become varying. Therefore, this study uses different notations to

present the capacities of passenger car and heavy vehicle. In addition, letting the unit of

capacity to be equal to the vehicle type is favorable to the derivative of the requirements of

the optimal solution of the proposed marginal-cost pricing model. Therefore, Eq. (1a) and Eq.

(1b) can be transferred as below (Yang and Huang, 2005):

(1c)

(1d)

where , and . Eq. (1c) is the cost function of passenger

car, which is not only affected by its own vehicle type, but also heavy vehicle flow where the

effect of a heavy vehicle is equal to times of the effect of a passenger car. Eq. (1d) is the

cost function of heavy vehicle, which is not only affected by its own vehicle type, but also

passenger car where the effect of a passenger car is equal to times of the effect of a heavy

vehicle. The degrees of the cross effect depend on the parameters of and

The demand function, which is nonnegative and strictly decreasing with the increasing

generalized cost, is assumed linear and is shown below:

(2a)

where is demand function of the m-th class between an O-D pair r-s; is class m’s

demand for travel between an O-D pair r-s; is class m’s potential demand for travel

between an O-D pair r-s; is the price elastic parameter of class ; is

generalized travel cost of class between an O-D pair r-s at traffic equilibrium. The

inverse demand function is presented as:

(2b)


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where is inverse of the demand function of the m-th class between an O-D

pair r-s. The inverse demand function can be regard as the willingness-to-pay

of user for taking his/her trip.

In an elastic demand case with asymmetric link cost function, the marginal-cost pricing

problem can be formulated as a Variational Inequality (VI) formulation as below:

(3a)

s.t. (3b)

(3c)

(3d)

(3e)

Eq. (3b) is the O-D demand conservation constraint for all classes of road users between all

O-D pairs. Eq. (3c) is the definitional equation, which represents link flows in terms of path

flows. Eq. (3d) and Eq. (3e) are the non-negativity constraints of the path flow and O-D

demand.

Based on the VI formulation, the marginal-cost pricing problem can be transformed to

mathematical programming problem as below:

(4)

subject to Eqs. (3b)-(3e) where is the link flows which include all vehicle types instead of

vehicle type m, on link a. Eq. (4) is the objective function to maximize the social benefit (SB).

The first term of the right-hand side is the total user benefit (UB) which is assumed dependent

on . The second term is the total travel cost (TC) which is assumed to be dependent on

, the flows on link a. The travel time costs of different users are the respective link

travel time multiplied by their own value of time, to become a monetary unit.

2.3 Uniqueness of Solution

In order to obtain the unique solution of link flows and tolls by using the current algorithm,

the objective function and the feasible region of solution must be convex. The concavity of

the feasible region is ensured by the linear equality constraints (Eq. (3b), Eq. (3d) and Eq. (3e))

(Shiffi, 1985). Different from the proof of the model developed by Yang and Huang (2005),

besides the asymmetric link cost function; this study also considers the heterogeneous value

of time. To guarantee the uniqueness of the solution, it needs to proof the concavity of the

objective function.

This study assumes that the demand function for each class of O-D pair, is

nonnegative and strictly decreasing with the increasing generalized cost for each class of users.

The inverse of the demand function, should be also a strictly decreasing function. The

integral of a decreasing function is strictly concave and sum of the strictly concave functions

is strictly concave. The negative of a strictly concave function will be convex. Sum of two

convex functions is convex. Therefore, we can focus on examining the convexity of the TC

with the asymmetric link cost function, which introduced in Eq. (1c) and Eq. (1d) to guarantee

the concavity of the objective function by insuring the positive definiteness of the Hessian

matrix. Specifically, this study sets:


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(5)

and

( )( )

(6)

According to Eq. (6), three conditions guarantee the positive definite of the Hessian matrix or

the convexity of the TC (Yang and Huang, 2005):

(i)

(ii)

(iii)

should be small.

The first and second requirements are the same as those discussed in (Yang and Huang,

2005).The third requirement would also be the same as Yang and Huang (2005) if this study

sets the VOT are equal to one. The third requirement means that the difference of asymmetric

effect in monetary unit between passenger car and heavy vehicle should be small to ensure the

negative value of second term of Eq. (6) to be smaller than the sum of other positive term.

The first and second requirements can be guaranteed by setting proper parameters. However,

the third requirement is hard to be guaranteed. We cannot set in advance or know the value of

because traffic flows are variable.

It causes the uncertainty of concavity of the objective function. Therefore, it may incur the

multiple solutions problem.


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2.4 Optimal Condition

The optimal conditions for the system optimum multiple vehicle types with elastic demand

can be derived by the first-order condition. Let link flow as a function of path flow vector f

defined by Eq. (3c) and the Lagrangian is presented as follows:

(7a)

s.t.

(7b)

(7c)

where are the dual variables (Lagrange multiplier) corresponding to Eq. (3b). The

first-order conditions can be written as:

(8a)

(8b)

(8c)

(8d)

(8e)

(8f)

(8g)

Here, this study calculates the partial derivatives of with respect to the flow

variable (which means flow of class users on path between the O-D pair n-o) and

the demand variable (which means class p’s demand for travel between the O-D pair

n-o). According to the derivation of Eq. (8a), we can know the congestion toll as follows:

(9)

where is the generalized cost (travel time cost and toll) of class on path


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between an O-D pair n-o. Here, is the external cost that a new

entry of vehicle type p imposes on other vehicle on link b. This term is the congestion toll

( ) that should be charged based on the marginal-cost pricing theory. Therefore, it can

present that:

(10)

The derivations of first-order conditions for this model are summarized as follows:

(11a)

(11b)

(11c)

(11d)

(11e)

(11f)

(11g)

Eqs. (11a) and (11b) are derived from Eqs. (8a) and (8b) and mean that if the cost of path k, is

larger than the minimum-path travel cost for the O-D pair r-s, no user will use this path and

the flow is zero. If the cost of path k is equal to the minimum-path travel cost, the flow on

path k can be zero or positive. Eqs. (11c) and (11d) are derived from Eqs. (8c) and (8d) and

mean that if the inverse of the demand function is smaller than the

minimum-path travel cost for the O-D pair r-s, the demand of O-D pair r-s is zero. If the

demand function is equal to the minimum-path travel cost, the demand of O-D

pair r-s can be zero or positive.

2.5 Frank-Wolfe Algorithm with Diagonalization Method

The diagonalization method is the most common algorithm to address the network assignment

problem with asymmetric link cost functions. This problem is similar as the network

assignment problem with single type of user, which can be solved by the Frank-Wolfe

algorithm. The difference is that the travel time on each link is updated based on the entire

flow pattern under different vehicle types, since rather than .To

deal with this problem, the link cost function Jacobian still needs to be positive definite to

assure a unique solution in the solving process. To satisfy the positive definite of link cost

function, this method only focuses on dealing with the main effect of a vehicle type on each

link (i.e. is a function of only). The cross effect of other vehicle type(s) on the same

link are fixed on every iterative.

This study combines the Frank-Wolfe algorithm and diagonalization method to solve the

problem. The problem will be mainly solved by the structure of the Frank-Wolfe algorithm.

The concept of diagonalization method is used to find the optimal step size to guarantee the

positive definite of link cost function.

In order to solve the problem by these algorithms, the formulations of this problem are

modified as follows:


395

(12a)

s.t.

(12b)

(12c)

(12d)

For computational reasons, is added as an upper bound constraint to the O-D

demand, . In general, is set to be the amount of potential demand. Due to:

(13a)

(13b)

The model can be further rewritten as:

(14a)

s.t.

(14b)

(14c)

In order to minimize the objective function by using the convex combinations algorithm, a

solution of the linear program is required at every iteration:

(15a)

s.t.

(15b)

(15c)

where is the auxiliary flow variable of class users on path

between an O-D pair r-s. A partial derivation of is presented as:

(16)

The program (15) can be expressed as:

(17a)

s.t.

(17b)

(17c)


396

where,

(17d)

(17e)

is the auxiliary flow variable of class on link . is the auxiliary O-D

flow variable of class between an O-D pair r-s. The program (17) can be solved

simply by inspection. To minimize , the demand for travel between an O-D pair r-s,

should be all assigned on the path, , for which and

is negative. If is positive, the

demand for the travel between an O-D pair r-s, should not be assigned on any path. According

to the above description, the minimization rule is presented as follows:

If (18a)

If (18b)

The SO formulation is generally not an equilibrium flow pattern, because the users can be

better off by changing their paths. Therefore, beside the travel cost that travelers perceived,

the external cost, , that travelers impose on the total system should be

charged by a toll. By considering the toll under a network equilibrium status, the model can

be expressed as follows:

(19a)

s.t.

(19b)

(19c)

To minimize the objective function along the descent direction(s), the moving size in the

Frank-Wolfe convex combinations method is determined by finding . The optimal step

size can be found by solving the following program:

(20a)

s.t.

(20b)

In order to ensure the link cost function Jacobian is positive definite, this study used the

diagonalization method. Program (20) only discusses the main effect by fixing the cross effect.


397

When the optimal moving size, , has been determined, and can be

updated by setting:

(21a)

(21b)

Convergence can be tested by the objective function values or the change in link flow and

demand between two successive iterations. Here, this study uses:

(22)

is the predetermined value that is based on the desired degree of accuracy.

According to the above description, the algorithmic steps of the solution algorithm can

be presented as follows:

Step 0: Initialization. Find an initial feasible flow pattern { }, { }.This study

assumes that there are no users in the network as initial feasible flow pattern and set

Step 1: Cost and benefit update. Set by using Eq. (3.1c) and

Eq. (3.1d); compute by using Eq. (3.2).

Step 2: Direction finding. Compute the shortest path, p, between each O-D pair r-s based on

{ }; execute the all-or-nothing assignment procedure according to Eq. (18) and then yield

an auxiliary flow pattern , by Eqs. (13).

Step 3: Moving-size determination. Find by solving program (20). This study uses the

golden section search to solve this program.

Step 4: Updating. Find and by solving Eq. (21).

Step 5: Convergence check. If inequality (22) is met, then terminate. Otherwise, set

and go to step 1.

3. CASE STUDY

In the case study, we aim to investigate the congestion pricing problem based on the National

Freeway No. 5 system in Taiwan. On the National Freeway No. 5 system, Hsuehshan Tunnel

which is one of the longest tunnels in the world (12.9 kilometers in length) and connects

Taipei metropolitan and Yilan county, which is one of the congested roadway segment in

Taiwan because the capacity of tunnel is merely 1,000 vehicles/phpl (Lin and Su, 2009).

Traffic congestion is more severe during weekends because many recreational trips are made

from Taipei to Yilan where many attractive spots are located.

We focus on the traffic congestion problem during peak periods on Hsuehshan Tunnel

where only passenger car and bus are allowed; truck and trailer are prohibited. There are two

lanes on each direction of the whole freeway No. 5 system. Although the parallel Taipei-Yilan

Provincial Highway No. 9 does not charge the toll, it is still included in the network because it

is the only alternative route. The simplified network is presented in Figure 1. Node 1, 2, 3 and

5 are Nangang, Shiding, Pinglin and Toucheng interchange. Node 4 is a dummy node to

distinguish the link of freeway and highway between Pinglin and Toucheng.


398

Figure 1. The simplified national freeway No. 5 network.

The empirical study data were collected based on 2014, which had implemented

differential pricing strategies. The field data of traffic flow collected from TANFB and

Directorate General of Highways is used to estimate demand function. Because the

differentiated tolls are not implemented widely and frequently in the freeway system, the

elastic parameters (and demand functions) are hard to be estimated by the property of vehicle

types, travel distance and so on. On the other hand, the travel demand of bus only accounts for

5.3% of total vehicles. Therefore, this study assumes the elastic parameters of bus are the

same as passenger car.

The related parameters of cost function include free flow travel time, capacity, value of

time and so on. The parameters of a BPR function ( and ) and capacity ( ) are

borrowed from the report of Institute of Transportation, MOTC (Lin et al., 2007). The

capacity unit of a bus is transformed from the passenger car unit (PCU) to the bus unit by

using passenger car equivalence (PCE), which is equal to two (Lin et al., 2007). The real

value of free flow travel time ( ) was obtained from TANFB. In addition, VOTs of

passenger car and bus are respectively set to be NT$ 8 and NT$ 20 per minute according to

the data of TANFB and Department of Statistics of MOTC.

3.1 Experimental Design and Results

In the case study, we design two scenarios to investigate the congestion tolls and compare the

resulted link/path flows with the real traffic flow distribution under a distance-based toll

scheme implemented in the field. The scenarios include vehicles with general lane and

vehicles with exclusive lane. The assumption of exclusive lane is because the bus exclusive

lane is implemented on a part of freeway No. 5 system (e.g., Yilan to Toucheng and a part of

Shiding to Pinglin). A possible policy will be implemented comprehensively in the future.

3.1.1 Vehicles without lane restriction (Scenario 1)

In this scenario, this study assumes that passenger car and bus can drive on each lane and they

can change lanes as needed. The parameters of the cross effects are set: and

(Chen and Bernstein, 2004).

According to the result, we found that the solution of the network equilibrium condition

is local optimum due to the non-concavity of the objective function. The reason is that, in this

scenario, the third requirement of the cross effect is too large that makes the objective

function be nonconvex.

Freeway system No. 5 Taipei-Yilan Provincial

Highway No. 9

Hsuehshan Tunnel

1 2 3 5

4


399

In this local optimal solution, according to the results shown in Figure 2 and Figure 3, we

can know that the tolls of passenger car and bus on the freeway segment are below the

standard toll, except for the freeway segment between node 3 (Pinglin) and node 5

(Toucheng). Both freeway and highway segments which connect node 3 (Pinglin) and node 5

(Toucheng) need to charge a huge amount of congestion toll which is above NT$ 100. The toll

of passenger car and bus on this freeway segment are thirteen and ten times higher than the

standard toll.

Figure 2. Ratio of passenger car toll between congestion toll and standard toll for scenario 1

Figure 3. Ratio of bus toll between congestion toll and standard toll for scenario 1

3.1.2 Vehicles with exclusive lane (Scenario 2)

In this scenario, this study assumes that the outside lane is a bus exclusive lane and passenger

car can only drive on the inside lane of the freeway segment. Since vehicles of different types

are traveling on specific lanes of the freeway No. 5 system, the cross effect on the freeway

segment is zero. On the highway segment, passenger car and bus are driving on the same

lanes. Therefore, the cross effect is assumed to be: γ=1 and δ=0.25.

Different from Scenario 1, the solution in this scenario is under a global optimum

condition. Every generalized path cost and O-D demand match the optimal conditions that

this study proved. As the demonstration in Figure 4, except for the link of node 2 (Shiding) to

node 3 (Pinglin), the toll levels of the other freeway segments should be increased by at least

twice of the standard toll. The toll of passenger car on this freeway segment of node 3

(Pinglin) and node 5 (Toucheng) are eighteen times higher than the standard toll. For the bus

(Figure 5), link tolls on the freeway segment are all below the standard toll levels due to the

bus exclusive lane policy, which is in favor of the transit mode when traveling on a congested

One to twice times

Below standard toll Twice to five times

Above five times

Cannot compare

1 2 3 5

4

One to twice times

Below standard toll

Twice to five times

Above five times

Cannot compare

1 2 3 5

4


400

tunnel section.

The tolls of passenger car and bus are different in this scenario due to the exclusive lane

control policy. The corresponding toll levels of bus are all equal to or less than that of

passenger car. The reason is that the exclusive lane policy is implemented on the freeway

system in this scenario. It reduces the available road capacity for passenger cars. In addition,

since the cross effect of bus is not existed, the toll of passenger car is increased and the toll of

bus is decreased.

Figure 4. Ratio of passenger car toll between congestion toll and standard toll for scenario 2

Figure 5. Ratio of bus toll between congestion toll and standard toll for scenario 2

3.2 Comparison of the Two Scenarios

In this section, this study compares the link flows of the two scenarios with real traffic flow

data. The data of real link flow and O-D demand is obtained from TANFB. It is the average

peak-hour flow of weekends between August 2014 and September 2014.

According to Figure 6 and Figure 7, the traffic flow of the Freeway No. 5 system

between Pinglin and Toucheng, which is the most congested roadway segment, reduced

significantly. In scenario 1, the traffic flows on the southward and northward directions reduce

about 40% and 30%, respectively. In scenario 2, the traffic flows on the southward and

northward directions reduce respectively about 60% and 55%. A part of the traffic flow

transfers to the alternative road (Taipei-Yilan Provincial Highway No. 9). In addition, a part of

the traffic demand is canceled due to the high congestion toll. This result demonstrates that

increasing congestion toll is an effective means to mitigate the congestion problem of the

Hsuehshan Tunnel freeway segment. Further, the traffic flows between Nangang and

Toucheng on the freeway segment are also smaller than that of the collected field data. The

traffic flows of field data between Nangang and Pinglin on the highway segment all transfers

to the corresponding freeway segment. Therefore, the flows of this highway segment become

zero in scenario 1 and scenario 2. In addition, charging congestion toll with exclusive lane can

One to twice times

Below standard toll Twice to five times

Above five times

Cannot compare

1 2 3 5

4

One to twice times

Below standard toll

Twice to five times

Above five times

Cannot compare

1 2 3 5

4


401

reduce more traffic flow than that without lane restriction.

2792.332912.89 2914.12

132.84233.71

2557.822430.16

1693.8

0

604.06

2060.15 1991.1

1092.4

0

702.99

0

500

1000

1500

2000

2500

3000

3500

Nangang–Shiding Shiding-Pinglin Pinglin-Toucheng Nangang –Pinglin Pinglin-Toucheng

Real data Scenario 1 Scenario 2

(PCE) freeway highway

Figure 6. Link flows of the real data and those of two scenarios of the southern bound

2571.82 2576.292439.56

132.84233.71

2659.61

2515.54

1746.77

0

649.11

2174.022008.02

1129.33

0

766.52

0

500

1000

1500

2000

2500

3000

Shiding-Nangang Pinglin-Shiding Toucheng-Pinglin Pinglin-Nangang Toucheng-Pinglin

Real data Scenario 1 Scenario 2

(PCE) freeway highway

Figure 7. Link flows of the real data and those of two scenarios of the northern bound

4. CONCLUSIONS AND RECOMMENDATIONS

This study investigates the differential congestion toll problem by the marginal cost pricing

theory under different vehicle types’ consideration. Different from the past research, this study

assumes that different vehicle types have distinct impacts on the others and values of time.

According to the outcome of the research, we can know that, for the freeway segment

between Pinglin and Toucheng, the tolls of both passenger car and bus need to be increased

more than ten times of the standard tolls on every scenario no matter on the southward or the


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northward direction, except for the toll of bus in scenario 2. Second, the traffic flow between

Pinglin and Toucheng, which is most congested road segment, are reduced significantly by

charging the congestion toll. Therefore, it can conclude that if the traffic congestion problem

on the freeway No. 5 system, the ceiling by law (twice of the standard toll) needs to be

relaxed when the marginal-cost pricing principle is applied in order to achieve system

optimum.

Besides the above research findings and conclusions, there are some future study

directions that are worthy of further investigation. First, this study uses the diagonalization

algorithm and Frank-Wolfe algorithm to solve this problem. However, the optimal solution

cannot always be obtained due to the uncertainty of concavity of the objective function. The

other solution algorithms need to be pursued to overcome this research limitation. In addition,

this study only considers the external cost of time loss. Incorporating other external costs into

a pricing model, such as air and/or noise pollution, pavement damage and traffic accident is

another important issue to make the road pricing policy more complete.

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A Marginal-cost Pricing Model for Transportation Networks